Indeed, one reason for using AES in counter mode is this random access, which among other things enables parallel encryption. The same strategy works with Chacha20.
I'm no expert of course so I don't even know if there's an AEAD that can bring you integrity on parts of the input; at least I know that minilock (https://github.com/kaepora/miniLock/blob/master/README.md#-m...) builds some kind of counter mode where each chunk is properly encrypted and has everything needed to check its integrity.
You can also just combine Salsa and HMAC.
It's true that you need to authenticate your data, but this is true for any cipher that you use.
It's a bad idea to implement your own cipher code, no matter what you're doing. If you're looking to include Salsa/ChaCha in an application, use Nacl, which refuses to give you unauthenticated ciphertext.